CHAPTER 19
REACTOR PHYSICS FOR LIQUID METAL REACTOR DESIGN*
The flexibility of liquid metal fuel systems is such that they range over
several different reactor categories . Liquid metal reactors may be designed
as fast, intermediate, or thermal systems, with either circulating or static
fuel systems . The reactor core components consist of a fuel carrier such as
molten bismuth or lead, and a moderator such as graphite or beryllium, if
the neutrons within the reactor core are to be thermalized . If the fuel is
stationary, a second fluid is required as the reactor coolant .
In the simplest system, a high-temperature liquid-metal solution or
slurry would be pumped through an externally moderated reactor core . For
such a reactor, the neutron physics problems would be similar to those of
aqueous homogeneous systems . The chief difference would lie in the neutron spectrum, which would be higher because of weaker moderation and
higher operating temperatures .
The liquid-metal system that has received the greatest emphasis to date
is of the heterogeneous, circulating fuel type . This reactor, known as the
Liquid Metal Fuel Reactor (LMFR), has as its fuel a dilute solution of
enriched uranium in liquid bismuth, and graphite is used as both moderator
and reflector . With 0 233 as the fuel and Th 232 as the fertile material, the
reactor can be designed as a thermal breeder . Consideration is restricted
here to this reactor type but, wherever possible, information of a general
nature is included .
19-1 .

L'_MMFR

PARAMETERS

19-1 .1 Cross sections . Most of the cross sections required for neutron
physics studies of the LMFR can be obtained from BN L-325 . The following exceptions should be noted . The 2200 in, sec value of the absorption
cross section of graphite is given as 3 .2 Âą 0 .2 mb . The best experimental
value however is 3 .6 mb after correcting for the presence of such impurities
as B, N 2, etc . Graphite of density 1 .65 to 1 .70 g1 en is obtainable with
an absorption cross section of about 4 mb, including impurities . Graphite
of density 1 .8 g; cm 3 or higher is becoming available, but the purity of
this high-density graphite has not been well established .
The 2200 in,'see value of the absorption cross section of Bi 209 is 32 + 2 mb .
Two isomeric states of Bi210 are formed, one of which decays by 0-ernission
with a half-life of 5 days into 11 0 210 .
*Contributed by J . Chernick, Brookhaven National Laboratory .

TABLE 19-1
PARAMETERS OF Br 209 RESONANCES

E o (ev)

v oI', barn-ev

810

9400

2370

7660

F, ev
5 .8 f 0 .3
f 1 .5

17

Bismuth has prominent resonances at 810 ev and 2370 ev, largely due
to scattering . Breit-Wigner parameters obtained by Bollinger et al . at
Argonne National Laboratory are listed in Table 19-1 . To determine neutron capture within these resonances, it is necessary to estimate the value
of the level width, F y . One method is to use the value of 0 .5 b obtained
by Langsdorf (ANL-4342) for the resonance integral, which implies that
F is about 150 mv . An analysis of Bollinger's data indicates that a more
likely value is about 50 mv .
High-energy cross sections of bismuth and lead are of secondary interest
in well-moderated liquid-metal reactors, but would become of prime interest in fast- or intermediate-energy reactors . On the basis of the known
levels arid spin assignments for bismuth and lead, Oleksa of Brookhaven
National Laboratory has calculated cross sections that are in good agreement with experimental data . The (n, p) and (n, a) cross sections are negligible . The threshold for the (n, 2n) cross section in bismuth is high,
7 .5 Mev . At 1 .0 and 4 .3 1M1ev the transport cross sections of bismuth are
calculated as 4 .3 b arid 4 .2 b, respectively . The capture cross section at
1 Mev is 3 .4 nib .
Inelastic scattering in bismuth is important in considering fission-energy
neutrons . The results of Oleksa's studies are presented in Table 19-2 .
The lowest levels in B1 209 occur at 0 .9, 1 .6, 3 .35 Mev, respectively . At
energies up to 2 .6 Mev, Oleksa finds that the cross sections for scattering
into the individual levels are in good agreement with calculations based
on the Hauser-Feshbach model .
In a U-fueled liquid-metal system, the cross sections of the higher isotopes or uranium are of considerable importance in determining equilibrium concentrations of these isotopes arid the time required to approach
their equilibrium . These equilibrium conditions require study because of
solubility limitations in a liquid-metal fuel reactor . The chain starts with
either U235 or U 233 depending on whether a converter or breeder reactor
is under consideration, and ends with U 237 because of its short half-life .
In addition, some U 238 may be present in the fuel . Thermal cross sections
are given in BNL-325 .

Other absorption cross sections of importance to high-power, high-fuelburziup reactors are those of the long-lived fission products and, in a U 233
breeder, that of Pa 233. Despite a number of comprehensive studies of these
effects, accurate values may not be known until such reactors have been
in operation for some time . Fuel-processing studies for the L,\1F R, however, indicate that the poisoning effect can economically be maintained at
a few percent .
Although the LMFR is a heterogeneous reactor, the fuel and moderator
arrangements that have been proposed yield a core which is nearly homogeneous from the neutron physics viewpoint . The preferred core is an
impermeable graphite structure perforated with holes of about 2 in . diameter for passage of the liquid-metal fuel . The moderator volume is about
equal to that of the liquid metal, bismuth, which contains about 0 .1 w/o
enriched uranium . Actually, the size of the fuel channels could be considerably increased without seriously increasing the flux disadvantage factor
and, hence, the critical mass of the reactor core .
19-1 .2 Neutron age and diffusion length . The following formulas, appropriate for mixtures, have been used to obtain the diffusion area, L 2 ,
of graphite-bismuth LMFR cores :
and neutron age,
T,

L2

(19 -1)

= 31a2;tr,
7-c(1

C

(~I,)Bi

+R)2

1 + (E1s)C R

,I
J

(~tr)Bi

C 1 + (ftr)C

,

R

( 19-2)

I

J

where ~ is the logarithmic energy decrement,
1, is the macroscopic scattering cross section,
is the macroscopic absorption cross section,
2:tr is the macroscopic transport cross section,
the subscripts Bi and C indicate the macroscopic cross section for
the respective materials, and R is the bismuth-to-graphite volume
ratio .
`a

19-1 .3 Reactivity effects . A problem unique to circulating fuel reactors
is the loss of delayed neutrons in the external circuit . Since the time spent
by the delayed-neutron emitters outside the reactor core is generally greater
than that spent within the core, a considerable fraction of the delayed neutrons may be wasted . In addition, since most of the delayed-neutron emitters are produced as gases, they may be carried off during degassing operations . For U233, the delayed neutron fraction in thermal fission is only
0.24% . Thus prompt critical may, in some cases, be as little as 0 .1%
excess reactivity .

TABLE 19-2
INELASTIC SCATTERING CROSS SECTION OF BI

E, Mev

v;1Bi, barns

0 .9
1 .0

0
0 .1

1 .5
2 .0
3 .0
4 .0
5 .0
6 .0
7 .0
8 .0
10 .0

0 .4
0 .7
1 .4
2 .0
2 .4
2 .6
2 .6
2 .5
1 .5

Coupled with this problem is the fact that the prompt temperature
coefficient (due to liquid metal expansion) in the LMFR system under
consideration is of the order of -5 X 10 - 5/°C . Thus, ignoring temperature
overshoots, which are discussed later, the magnitude of rapid reactivity
changes must be limited to avoid large metal temperature changes . The
total temperature coefficient of the LMFR runs about -1 .5 X 10-4/°C,
the delayed coefficient resulting primarily from increased neutron leakage
due to the heating of the graphite structure . While the slow response of
the graphite to power changes thus limits the size of the prompt temperature coefficient, it aids in stabilizing the system against small oscillations
at high power output .
19-1 .4 Breeding . The LMFR can be operated as a breeder on the
U 233-Th232 cycle . The possible breeding gain is not large, since the value
of 77 for 0 233 is about 2 .3 . The theoretical gain is at most 0 .3, but a value
of 0 .10 is about the maximum possible in a practical system. In fact,
optimization based on economic considerations would probably reduce the
gain to zero in any power breeder built in the near future . The gain is
reduced by competitive neutron capture in the core and blanket, and by
neutron leakage from the blanket and from the ends of the reactor core .
A problem not yet solved is that of a leakproof, weakly absorbing container that will separate the core and blanket . It is hoped that beryllium
or an impermeable graphite will provide such a container for the LMFR •
Croloy steel or tantalum containers about 1/4-in thick appear satisfactory

from the mechanical and metallurgical standpoint but effectively wipe out
the potential breeding gain because of their absorption cross section .
A number of studies of the so-called immoderation principle have been
carried out in an attempt to reduce the neutron losses to the container .
By removing the bulk of the moderator from a small region on both sides
of the container wall the thermal neutron losses can be greatly reduced .
Several feasible mechanical designs embodying this principle have been
worked out by the Babcock & Wilcox Company .
19-2 . LMFR STATICS

19-2 .1 Core . The standard LMFR is predominantly thermal, nearly
homogeneous, and moderated by graphite . Thus age-diffusion theory is
applicable, and therefore the following formula can be used for a critical
system
k,,e -,B'

= 1 + L 2B2,

( 19-3)

where B 2 is the buckling of the system, and
k . = -qf,

(19-4)

where 77 is the number of fast neutrons produced per thermal neutron
captured in the fuel, and f is the thermal utilization factor . The product
of the fast fission effect, e, and resonance escape probability, p, is assumed
equal to unity .
In view of the uncertainty in the value of F, for bismuth, the validity of
neglecting resonance capture is still uncertain, and Monte Carlo studies
are planned at B7',-,I, to obtain lower limits for p as a function of channel
size and lattice pitch . For small channels, the homogeneous formula for f
is adequate, since consideration of self-shielding of the fuel reduces f in a
typical core by about 2% .
Studies have yielded for buckling the typical values given in Table 19-3 [1]
for both 1. 233 and U235 as the fuel in a graphite moderator at an average
core temperature of 4M 째 C.
19-2 .2 Reflector. In order to apply the above results to reflected reactors, it is necessary to determine the reflector savings, which can be
obtained from conventional two-group theory . This method could also
be used to estimate the critical size of the reactor but, for small cores, twogroup theory underestimates the size of graphite-moderated reactors .
Two-group results obtained for typical reflectors are given in Table
19-4 [1] for a cylindrical reactor system surrounded by a large reflector .

19-2 .3 Critical mass . The results of age-diffusion theory are in good
agreement with multigroup calculations for predominantly thermal LMFR
reactors . At higher fuel concentrations, however, the age theory overestimates the critical mass, as shown in Table 19-5 [1] . The differences in
critical mass estimates are large only for weakly moderated reactors .
TABLE 19-5
CRITICAL MASS AND DIAMETER OF U 235-FUELED

LMFR

SPHERES WITH A 90-CM GRAPHITE REFLECTOR

Age-Diffusion
VU// \fBi

Multigroup

VBi U/I'C

Diameter, ft

Mass, kg

Diameter, ft

Mass, kg

4 .52
3 .81
4 .04
2 .28
2 .24

3 .02
4 .53
7 .15
3 .89
9 .20

3 .88
2 .94
3 .04
1 .66
1 .73
1 .94

1 .92
2 .07
3 .05
1 .49
4 .19
7 .87

Graphite-moderated
x 10 -3

0 .25
1 .0
2 .0

1 x 10 -2

0 .25
1 .0

1

4 .38
3 .88
4 .15
2 .57
2 .79

2 .73
4 .77
7 .79
5 .50
17 .69

Beryllium-moderated
I x 10 -3

0 .25
1 .0
2 .0

1 x 10-2

0 .25
1 .0
2 .0

3 .86
3 .08
3 .29
1 .90
2 .11
2 .43

1 .87
2 .37
3 .86
2 .22
7 .70
15 .55

19-2 .4 Breeding. The conversion ratio obtainable in liquid metal systems depends on a number of variables, such as the fuel and fertile material
concentrations, the fission-product processing methods, losses to the core
container, etc . In a feasibility study of the LMFR conducted by the
Babcock & Wilcox Company, currently practical reactor designs were
reported (BAW-2) with conversion ratios ranging from 0.8 to 0 .9, depending on whether an oxide slagging or fused salt method was used for
nonvolatile fission-product processing . The U/Bi atomic ratio was low
(0 .6 X 10 -3 ) and a 2'% Cr-1% Mo steel core container was used, both

choices tending to reduce the possible breeding ratio . The estimates of
the neutron balance are given in Table 19-6 [6] .
TABLE

19-6

NEUTRON BALANCE OF TH232

Production per U233 absorption

U233

BREEDER

Scheme A
Oxide slagging

Scheme B
Fused-salt process

2 .31

2 .31

1 .00
0 .13
0 .05
0 .12
0 .02
0 .12
0 .80
0 .02
0 .05

1 .00
0 .13
0 .05
0 .03
0 .02
0 .12
0 .89
0 .02
0 .05

Losses : Absorption in
U233

Bi
C

Fission products
Higher isotopes
Croloy structure
Th

Pa
Leakage

19-2 .5 Control . Because of its prompt temperature coefficient, the
LMFR is expected to be stable . Nevertheless, it represents a completely
new and untested system . There are a number of ways in which the
reactivity of the system can change, for example, with changes in inlet
temperature, concentration, or velocity of the fuel, and changes in xenon
concentration, delayed neutron emitter concentration, and blanket composition . Most of these changes are expected to be gradual, but they can
be sufficiently large to require the use of control rods . Inherent stability
has not been demonstrated in operating reactors except over a limited
range in reactivity and power output . In a reactor with a high-velocity
coolant there may occur sudden changes of reactivity which are too fast
for conventional control. Thus both inherent stability against sudden
reactivity changes and control rods for large but gradual reactivity changes
are needed until considerable experience has been gained in operation of
the reactor .
Studies have been carried out at BNL on control requirements for an
LMFR experiment . The control requirements depend not only on the
choice of operating temperatures, the possible xenon and fission-product
poisoning, etc ., but also on conceivable emergency situations such as errors
in fuel concentration control . In a reactor with a full breeding blanket, the
control requirements may have to include the effect of complete loss of
the breeder fluid .

For a 5-mw experiment, control of 15% reactivity appears to be ample
and can be obtained with four 24% Cr-1% Mo steel rods of about 2-in .
diameter. Blacker rods containing boron could, of course, be used to increase reactivity control . A study of various arrangements of identical
rods in a ring around a central rod indicates that the optimum position of
the ring occurs at about 1/4 of the distance from the reactor center to the
(extrapolated) radius of the reactor core .
It would be highly desirable to use sheaths for control rods in order to
eliminate the problem of rod insertion through a heavy liquid metal . Steel
sheaths are not satisfactory, since they reduce the breeding ratio in a
liquid-metal power breeder and reduce the over-all thermal flux in an
experimental reactor . The solution to the problem may lie in the development of structurally sound beryllium sheaths .
19-2 .6 Shielding . Shielding of an LMFR is complicated by the necessity
of shielding an external circuit in which the delayed neutron emitters and
fission products decay .
Calculations by K . Spinney at BNL indicate that even for a 5-Mw
experimental reactor, about 5 .5 ft of concrete are required as a neutron
shield around the reactor cell . Gamma shielding of the cell requires about
8 .5 ft of ordinary concrete or 4 .5 ft of BNL concrete (70% Fe) . For this
reason, it has been proposed that heavy concrete be used as the shield for
the 5-Mw reactor . For the rest of the circuit, including the degasser,
pumps, heat exchanger, etc ., the advantage of using BNL concrete is
less evident .
19-3 . LMFR KINETICS
A number of fundamental studies of the kinetics of circulating fuel
reactors have been carried out at ORXL and by Babcock & Wilcox Company . A review of the subject has been given by Welton [2] . At low power,
the equations governing the system are linear and complicated chiefly by
the feedback of delayed neutrons . General results for the in-hour relation
have been obtained by Fleck [3] for U233- and IU235-fueled reactors . At
high power, the kinetics are much more complicated and there is a real
question whether the response of a complex reactor can be accurately predicted in advance of its operation . Bethe [6] has strongly recommended
the use of oscillator experiments to determine reactor transfer functions .
Despite such experiments, however, the mechanism responsible for the
resonances observed in EBR-1 has, to date, not been satisfactorily explained .
There are two methods of treating the kinetics of a reactor . In the openloop method, the inlet temperature is taken as constant . The justification
for this procedure is that this condition generally prevails during rapid

transients, the feedback of information through the external system being
slow by comparison . The method, however, suffers from the defect that it
cannot reveal instabilities associated with the entire circuit . In the closedloop method, the external system, or a reasonable facsimile, is coupled to
the reactor system . The representation of the reactor, however, is generally
oversimplified because of the complexity of the over-all system .
Although the set of kinetic equations that include temperature effects
are nonlinear, the linearized equations are satisfactory for the investigation
of stability and the qualitative transient behavior . A large subset of equations is required to properly treat the effect of the delayed neutron emitters .
Again, however, lumping the delayed neutrons into a single group, or
neglecting them altogether, always appears to lead to qualitatively, if not
quantitatively, correct results .
A study of the temperature-dependent open-loop kinetics of the LMFR
has been carried out by Fleck [4] . The effect of delayed neutrons and the
delayed moderator temperature coefficient were neglected . Under these
conditions, Fleck found that the reactor responded rapidly and with little
overshoot in temperature when subjected to the largest permissible reactivity excursions .
Using a method developed at the Oak Ridge National Laboratory
(ORNL-CFI-56-4-183) for homogeneous systems, the Babcock & Wilcox
Company has studied the stability of the LMFR against small oscillations .
The results show that the LMFR models under study are stable up to
power densities 100 to 1000 times greater than the nominal design level .
Fleck has also examined the transient pressures in LMFR cores by treating the bismuth as a frictionless, compressible fluid . He found that the
maximum pressures developed during conceivable transients were quite
small . The assumption sometimes made, that the fluid external to the core
can be represented as an incompressible slug, was found to overestimate
the transient pressures .
In general, heterogeneous reactors possessing both a small prompt
(positive or negative) fuel temperature coefficient and a large delayed
negative moderator temperature coefficient can be expected to exhibit
oscillatory instability at sufficiently high power . However, elementary
models indicate that power levels high enough to cause such instability
are not achievable in present reactors . Further study of the complex heattransfer transients in reactor systems is still required before reactor stability can be assured .